Calibrations for minimal networks in a covering space setting

In this paper we define a notion of calibration for an equivalent approach to the classical Steiner problem in a covering space setting and we give some explicit examples. Moreover we introduce the notion of calibration in families: the idea is to divide the set of competitors in a suitable way, defining an appropriate (and weaker) notion of calibration. Then, calibrating the candidate minimizers in each family and comparing their perimeter, it is possible to find the minimizers of the minimization problem. Thanks to this procedure we prove the minimality of the Steiner configurations spanning the vertices of a regular hexagon and of a regular pentagon

Neocortical Axon Arbors Trade-off Material and Conduction Delay Conservation

The brain contains a complex network of axons rapidly communicating information between billions of synaptically connected neurons. The morphology of individual axons, therefore, defines the course of information flow within the brain. More than a century ago, Ramón y Cajal proposed that conservation laws to save material (wire) length and limit conduction delay regulate the design of individual axon arbors in cerebral cortex. Yet the spatial and temporal communication costs of single neocortical axons remain undefined. Here, using reconstructions of in vivo labelled excitatory spiny cell and inhibitory basket cell intracortical axons combined with a variety of graph optimization algorithms, we empirically investigated Cajal's conservation laws in cerebral cortex for whole three-dimensional (3D) axon arbors, to our knowledge the first study of its kind. We found intracortical axons were significantly longer than optimal. The temporal cost of cortical axons was also suboptimal though far superior to wire-minimized arbors. We discovered that cortical axon branching appears to promote a low temporal dispersion of axonal latencies and a tight relationship between cortical distance and axonal latency. In addition, inhibitory basket cell axonal latencies may occur within a much narrower temporal window than excitatory spiny cell axons, which may help boost signal detection. Thus, to optimize neuronal network communication we find that a modest excess of axonal wire is traded-off to enhance arbor temporal economy and precision. Our results offer insight into the principles of brain organization and communication in and development of grey matter, where temporal precision is a crucial prerequisite for coincidence detection, synchronization and rapid network oscillations

Toward Controllable and Robust Surface Reconstruction from Spatial Curves

Reconstructing surface from a set of spatial curves is a fundamental problem in computer graphics and computational geometry. It often arises in many applications across various disciplines, such as industrial prototyping, artistic design and biomedical imaging. While the problem has been widely studied for years, challenges remain for handling different type of curve inputs while satisfying various constraints. We study studied three related computational tasks in this thesis. First, we propose an algorithm for reconstructing multi-labeled material interfaces from cross-sectional curves that allows for explicit topology control. Second, we addressed the consistency restoration, a critical but overlooked problem in applying algorithms of surface reconstruction to real-world cross-sections data. Lastly, we propose the Variational Implicit Point Set Surface which allows us to robustly handle noisy, sparse and non-uniform inputs, such as samples from spatial curves

Local Search Approximation Algorithms for Clustering Problems

In this research we study the use of local search in the design of approximation algorithms for NP-hard optimization problems. For our study we have selected several well-known clustering problems: k-facility location problem, minimum mutliway cut problem, and constrained maximum k-cut problem. We show that by careful use of the local optimality property of the solutions produced by local search algorithms it is possible to bound the ratio between solutions produced by local search approximation algorithms and optimum solutions. This ratio is known as the locality gap. The locality gap of our algorithm for the k-uncapacitated facility location problem is 2+sqrt(3) +epsilon for any constant epsilon \u3e0. This matches the approximation ratio of the best known algorithm for the problem, proposed by Zhang but our algorithm is simpler. For the minimum multiway cut problem our algorithm has locality gap 2-2/k, which matches the approximation ratio of the isolation heuristic of Dahlhaus et al; however, our experimental results show that in practice our local search algorithm greatly outperforms the isolation heuristic, and furthermore it has comparable performance as that of the three currently best algorithms for the minimum multiway cut problem. For the constrained maximum k-cut problem on hypergraphs we proposed a local search based approximation algorithm with locality gap 1-1/k for a variety of constraints imposed on the k-cuts. The locality gap of our algorithm matches the approximation ratio of the best known algorithm for the max k-cut problem on graphs designed by Vazirani, but our algorithm is more general

Surface-guided computing to analyze subcellular morphology and membrane-associated signals in 3D

Signal transduction and cell function are governed by the spatiotemporal organization of membrane-associated molecules. Despite significant advances in visualizing molecular distributions by 3D light microscopy, cell biologists still have limited quantitative understanding of the processes implicated in the regulation of molecular signals at the whole cell scale. In particular, complex and transient cell surface morphologies challenge the complete sampling of cell geometry, membrane-associated molecular concentration and activity and the computing of meaningful parameters such as the cofluctuation between morphology and signals. Here, we introduce u-Unwrap3D, a framework to remap arbitrarily complex 3D cell surfaces and membrane-associated signals into equivalent lower dimensional representations. The mappings are bidirectional, allowing the application of image processing operations in the data representation best suited for the task and to subsequently present the results in any of the other representations, including the original 3D cell surface. Leveraging this surface-guided computing paradigm, we track segmented surface motifs in 2D to quantify the recruitment of Septin polymers by blebbing events; we quantify actin enrichment in peripheral ruffles; and we measure the speed of ruffle movement along topographically complex cell surfaces. Thus, u-Unwrap3D provides access to spatiotemporal analyses of cell biological parameters on unconstrained 3D surface geometries and signals.Comment: 49 pages, 10 figure

Statistical mechanics approaches to optimization and inference

Nowadays, typical methodologies employed in statistical physics are successfully applied to a huge set of problems arising from different research fields. In this thesis I will propose several statistical mechanics based models able to deal with two types of problems: optimization and inference problems. The intrinsic difficulty that characterizes both problems is that, due to the hard combinatorial nature of optimization and inference, finding exact solutions would require hard and impractical computations. In fact, the time needed to perform these calculations, in almost all cases, scales exponentially with respect to relevant parameters of the system and thus cannot be accomplished in practice. As combinatorial optimization addresses the problem of finding a fair configuration of variables able to minimize/maximize an objective function, inference seeks a posteriori the most fair assignment of a set of variables given a partial knowledge of the system. These two problems can be re-phrased in a statistical mechanics framework where elementary components of a physical system interact according to the constraints of the original problem. The information at our disposal can be encoded in the Boltzmann distribution of the new variables which, if properly investigated, can provide the solutions to the original problems. As a consequence, the methodologies originally adopted in statistical mechanics to study and, eventually, approximate the Boltzmann distribution can be fruitfully applied for solving inference and optimization problems. The structure of the thesis follows the path covered during the three years of my Ph.D. At first, I will propose a set of combinatorial optimization problems on graphs, the Prize collecting and the Packing of Steiner trees problems. The tools used to face these hard problems rely on the zero-temperature implementation of the Belief Propagation algorithm, called Max Sum algorithm. The second set of problems proposed in this thesis falls under the name of linear estimation problems. One of them, the compressed sensing problem, will guide us in the modelling of these problems within a Bayesian framework along with the introduction of a powerful algorithm known as Expectation Propagation or Expectation Consistent in statistical physics. I will propose a similar approach to other challenging problems: the inference of metabolic fluxes, the inverse problem of the electro-encephalography and the reconstruction of tomographic images

Computerized Analysis of Magnetic Resonance Images to Study Cerebral Anatomy in Developing Neonates

The study of cerebral anatomy in developing neonates is of great importance for the understanding of brain development during the early period of life. This dissertation therefore focuses on three challenges in the modelling of cerebral anatomy in neonates during brain development. The methods that have been developed all use Magnetic Resonance Images (MRI) as source data. To facilitate study of vascular development in the neonatal period, a set of image analysis algorithms are developed to automatically extract and model cerebral vessel trees. The whole process consists of cerebral vessel tracking from automatically placed seed points, vessel tree generation, and vasculature registration and matching. These algorithms have been tested on clinical Time-of- Flight (TOF) MR angiographic datasets. To facilitate study of the neonatal cortex a complete cerebral cortex segmentation and reconstruction pipeline has been developed. Segmentation of the neonatal cortex is not effectively done by existing algorithms designed for the adult brain because the contrast between grey and white matter is reversed. This causes pixels containing tissue mixtures to be incorrectly labelled by conventional methods. The neonatal cortical segmentation method that has been developed is based on a novel expectation-maximization (EM) method with explicit correction for mislabelled partial volume voxels. Based on the resulting cortical segmentation, an implicit surface evolution technique is adopted for the reconstruction of the cortex in neonates. The performance of the method is investigated by performing a detailed landmark study. To facilitate study of cortical development, a cortical surface registration algorithm for aligning the cortical surface is developed. The method first inflates extracted cortical surfaces and then performs a non-rigid surface registration using free-form deformations (FFDs) to remove residual alignment. Validation experiments using data labelled by an expert observer demonstrate that the method can capture local changes and follow the growth of specific sulcus

Optimization based methods for solving some problems in telecommunications and the internet

The purpose of this thesis is to develop some new algorithms based on optimization techniques for solving some problems in some areas of telecommunications and the Internet. There are two main parts to this thesis. In the first part we discuss optimization based stochastic and queueing models in telecommunications network corrective maintenance. In the second part we develop optimization based clustering (OBC) algorithms for network evolution and multicast routing. The most typical scenario encountered during mathematical optimization modelling in telecommunications, for example, is to minimize the cost of establishment and maintenance of the networks subject to the performance constraints of the networks and the reliability constraints of the networks as well. Most of these optimization problems are global optimization, that is, they have many local minima and most of these local minima do not provide any useful information for solving these problems. Therefore, the development of effective methods for solving such global optimization problems is important. To run the telecommunications networks with cost-effective network maintenance,we need to establish a practical maintenance model and optimize it. In the first part of the thesis, we solve a known stochastic programming maintenance optimization model with a direct method and then develop some new models. After that we introduce queue programming models in telecommunications network maintenance optimization. The ideas of profit, loss, and penalty will help telecommunications companies have a good view of their maintenance policies and help them improve their service. In the second part of this thesis we propose the use of optimization based clustering (OBC) algorithms to determine level-constrained hierarchical trees for network evolution and multicast routing. This problem is formulated as an optimization problem with a non-smooth, non-convex objective function. Different algorithms are examined for solving this problem. Results of numerical experiments using some artifiicial and real-world databases are reported.Doctor of Philosoph

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor